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The billiard in an ellipse has an integral linear in momentum, theJoachimsthal integral. We show that the existence of such an integralcharacterizes conics. We extend this result to the spherical and hyperbolicgeometries and to higher dimensions. We connect the existence of Joachimsthalintegral with the Poritsky property, a property of billiard curves, called soafter H. Poritsky whose important 1950 paper was one of the early studies ofthe billiard problem.

Remarks on Joachimsthal integral and Poritsky property